AI 中文总结
本文修正了群表示加性直径的一个命题，并证明了加性直径在群与其李代数之间的不等式，无需不可约或连通假设。
AI 中文摘要
我们记录了关于群表示加性直径的两个结果。首先，我们给出了arXiv:2504.07573第一个arXiv版本中命题5.3的修正版本。设$0<\varepsilon<1/3$，$n>9/\varepsilon^2$，且$U\leq\mathfrak{sl}_n(\mathbb{C})$满足$\dim U>\varepsilon n^2$。则$\operatorname{diam}^{\mathrm{SL}_n(\mathbb{C})}_{+}(\mathfrak{sl}_n(\mathbb{C}),U)\leq 32/\varepsilon+8$。证明使用了对称群在非对角矩阵位置上的作用的平均化论证。该论证随后在arXiv:2609.03882中被进一步发展。我们还证明了arXiv:2504.07573中问题6.7的右侧不等式。若$G$是复代数群，$V$是有限维$G$-模，$U\leq V$，且$\mathfrak g=\operatorname{Lie}(G)$，则$\operatorname{diam}^{G}_{+}(V,U)\leq\operatorname{diam}^{\mathfrak g}_{+}(V,U)$。不需要不可约性或连通性假设。
英文摘要
We record two results on additive diameters of group representations. First we give a corrected version of Proposition 5.3 from the first arXiv version of arXiv:2504.07573. Let $0<\varepsilon<1/3$, let $n>9/\varepsilon^2$, and let $U\leq\mathfrak{sl}_n(\mathbb{C})$ with $\dim U>\varepsilon n^2$. Then $\operatorname{diam}^{\mathrm{SL}_n(\mathbb{C})}_{+}(\mathfrak{sl}_n(\mathbb{C}),U)\leq 32/\varepsilon+8$. The proof uses an averaging argument for the action of the symmetric group on the off-diagonal matrix positions. This argument was subsequently developed further in arXiv:2609.03882. We also prove the right-hand inequality in Question 6.7 of arXiv:2504.07573. If $G$ is a complex algebraic group, $V$ is a finite-dimensional $G$-module, $U\leq V$, and $\mathfrak g=\operatorname{Lie}(G)$, then $\operatorname{diam}^{G}_{+}(V,U)\leq\operatorname{diam}^{\mathfrak g}_{+}(V,U)$. No irreducibility or connectedness assumption is needed.
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